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A342723
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a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.
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3
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0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 3, 1, 2, 0, 0, 4, 3, 0, 0, 4, 7, 2, 0, 5, 2, 5, 0, 1, 0, 3, 4, 3, 4, 0, 6, 7, 3, 4, 0, 3, 4, 0, 0, 5, 0, 9, 10, 9, 3, 0, 5, 8, 0, 4, 0, 17, 4, 0, 9, 1, 19, 2, 0, 6, 2, 7, 0, 7, 7, 7, 23, 2, 8, 12, 0, 10, 0, 5, 0, 15, 27
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OFFSET
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1,12
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COMMENTS
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Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.
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LINKS
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EXAMPLE
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a(4)=1 is the smallest possible solution and is a rectangle with a=c=4, b=d=3, e=f=5 and area 12. a(24)=4 includes the smallest possible solution with all sides a,b,c,d different and a=24, b=20, c=15, d=7, e=20, f=25 and area 234. Furthermore there are three rectangles with a=24,b=7, a=24,b=10 and a=24,b=18.
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MATHEMATICA
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an={};
area[a_, b_, c_, d_, e_, f_]:=1/4 Sqrt[4e^2 f^2-(a^2+c^2-b^2-d^2)^2];
he[a_, b_, e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
paX[e_]:={e, 0} (*vertex A coordinate*)
pbX[a_, b_, e_]:={(-a^2+b^2+e^2)/(2 e), he[a, b, e]}(*vertex B coordinate*)
pc={0, 0}; (*vertex C coordinate*)pdX[c_, d_, e_]:={(c^2-d^2+e^2)/(2 e), -he[c, d, e]}(*vertex D coordinate*)
convexQ[{bx_, by_}, {dx_, dy_}, e_]:=If[(by-dy) e>by dx-bx dy>0, True, False]
(*define order on tuples*)
gQ[x_, y_]:=Module[{z=x-y, res=False}, Do[If[z[[i]]>0, res=True; Break[], If[z[[i]]<0, Break[]]], {i, 1, 6}]; res]
(*check if tuple is canonical*)
canonicalQ[{a_, b_, c_, d_, e_, f_}]:=Module[{x={a, b, c, d, e, f}}, If[(gQ[{b, a, d, c, e, f}, x]||gQ[{d, c, b, a, e, f}, x]||gQ[{c, d, a, b, e, f}, x]||gQ[{b, c, d, a, f, e}, x]||gQ[{a, d, c, b, f, e}, x]||gQ[{c, b, a, d, f, e}, x]||gQ[{d, a, b, c, f, e}, x]), False, True]]
Do[cnt=0;
Do[pa=paX[e]; pb=pbX[a, b, e]; pd=pdX[c, d, e];
If[(f=Sqrt[(pb-pd).(pb-pd)]; IntegerQ[f])&&(ar=area[a, b, c, d, e, f]; IntegerQ[ar])&&convexQ[pb, pd, e]&&canonicalQ[{a, b, c, d, e, f}], cnt++
(*; Print[{{a, b, c, d, e, f, ar}, Graphics[Line[{pa, pb, pc, pd, pa}]]}]*)], {b, 1, a}, {e, a-b+1, a+b-1}, {c, 1, a}, {d, Abs[e-c]+1, Min[a, e+c-1]}];
AppendTo[an, cnt], {a, 1, 85}]
an
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CROSSREFS
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Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342722 for convex integer quadrilaterals with arbitrary area.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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